Although there has been significant success with this approach, there are several ways to improve the performance of fixed-boundary optimization. Specifically, the incorporation of derivative information could be transformative. Fixed-boundary optimization has previously relied on derivative-free methods – such as genetic algorithms [1]}, differential evolution [2]}, and Brent's algorithm [3]} – or derivative-based methods with finite-difference gradients. While global derivative-free algorithms may prevent the optimization from terminating in local minima, they are only effective for smaller problems [4]}. Gradient-based optimization with finite-difference gradients requires excessive function evaluations in high-dimensional spaces and suffers from error that enters due to the choice of step size. If the step size is too small, the error is dominated by round-off error, and if it is too large it is dominated by nonlinearity [5]}. Because of the requirement of excessive function evaluations and the unreliability of the gradient information, gradient-based optimization with finite-difference derivatives is not always effective. In this work, we present the first example of analytic gradient-based, fixed-boundary optimization of stellarator equilibria.
